Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  acexmidlemcase Unicode version

Theorem acexmidlemcase 5538
 Description: Lemma for acexmid 5542. Here we divide the proof into cases (based on the disjunction implicit in an unordered pair, not the sort of case elimination which relies on excluded middle). The cases are (1) the choice function evaluated at equals , (2) the choice function evaluated at equals , and (3) the choice function evaluated at equals and the choice function evaluated at equals . Because of the way we represent the choice function , the choice function evaluated at is and the choice function evaluated at is . Other than the difference in notation these work just as and would if were a function as defined by df-fun 4934. Although it isn't exactly about the division into cases, it is also convenient for this lemma to also include the step that if the choice function evaluated at equals , then and likewise for . (Contributed by Jim Kingdon, 7-Aug-2019.)
Hypotheses
Ref Expression
acexmidlem.a
acexmidlem.b
acexmidlem.c
Assertion
Ref Expression
acexmidlemcase
Distinct variable groups:   ,,,,,   ,,,,,   ,,,,,   ,,,,,

Proof of Theorem acexmidlemcase
StepHypRef Expression
1 acexmidlem.a . . . . . . . . . . . . . 14
2 onsucelsucexmidlem 4280 . . . . . . . . . . . . . 14
31, 2eqeltri 2152 . . . . . . . . . . . . 13
4 prid1g 3504 . . . . . . . . . . . . 13
53, 4ax-mp 7 . . . . . . . . . . . 12
6 acexmidlem.c . . . . . . . . . . . 12
75, 6eleqtrri 2155 . . . . . . . . . . 11
8 eleq1 2142 . . . . . . . . . . . . . . 15
98anbi1d 453 . . . . . . . . . . . . . 14
109rexbidv 2370 . . . . . . . . . . . . 13
1110reueqd 2560 . . . . . . . . . . . 12
1211rspcv 2698 . . . . . . . . . . 11
137, 12ax-mp 7 . . . . . . . . . 10
14 riotacl 5513 . . . . . . . . . 10
1513, 14syl 14 . . . . . . . . 9
16 elrabi 2747 . . . . . . . . . 10
1716, 1eleq2s 2174 . . . . . . . . 9
18 elpri 3429 . . . . . . . . 9
1915, 17, 183syl 17 . . . . . . . 8
20 eleq1 2142 . . . . . . . . . 10
2115, 20syl5ibcom 153 . . . . . . . . 9
2221orim2d 735 . . . . . . . 8
2319, 22mpd 13 . . . . . . 7
24 acexmidlem.b . . . . . . . . . . . . . 14
25 pp0ex 3968 . . . . . . . . . . . . . . 15
2625rabex 3930 . . . . . . . . . . . . . 14
2724, 26eqeltri 2152 . . . . . . . . . . . . 13
2827prid2 3507 . . . . . . . . . . . 12
2928, 6eleqtrri 2155 . . . . . . . . . . 11
30 eleq1 2142 . . . . . . . . . . . . . . 15
3130anbi1d 453 . . . . . . . . . . . . . 14
3231rexbidv 2370 . . . . . . . . . . . . 13
3332reueqd 2560 . . . . . . . . . . . 12
3433rspcv 2698 . . . . . . . . . . 11
3529, 34ax-mp 7 . . . . . . . . . 10
36 riotacl 5513 . . . . . . . . . 10
3735, 36syl 14 . . . . . . . . 9
38 elrabi 2747 . . . . . . . . . 10
3938, 24eleq2s 2174 . . . . . . . . 9
40 elpri 3429 . . . . . . . . 9
4137, 39, 403syl 17 . . . . . . . 8
42 eleq1 2142 . . . . . . . . . 10
4337, 42syl5ibcom 153 . . . . . . . . 9
4443orim1d 734 . . . . . . . 8
4541, 44mpd 13 . . . . . . 7
4623, 45jca 300 . . . . . 6
47 anddi 768 . . . . . 6
4846, 47sylib 120 . . . . 5
49 simpl 107 . . . . . . 7
50 simpl 107 . . . . . . 7
5149, 50jaoi 669 . . . . . 6
5251orim2i 711 . . . . 5
5348, 52syl 14 . . . 4
5453orcomd 681 . . 3
55 simpr 108 . . . . 5
5655orim1i 710 . . . 4
5756orim2i 711 . . 3
5854, 57syl 14 . 2
59 3orass 923 . 2
6058, 59sylibr 132 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   wo 662   w3o 919   wceq 1285   wcel 1434  wral 2349  wrex 2350  wreu 2351  crab 2353  cvv 2602  c0 3258  csn 3406  cpr 3407  con0 4126  crio 5498 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-nul 3912  ax-pow 3956 This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-uni 3610  df-tr 3884  df-iord 4129  df-on 4131  df-suc 4134  df-iota 4897  df-riota 5499 This theorem is referenced by:  acexmidlem1  5539
 Copyright terms: Public domain W3C validator